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Superreplication in Stochastic Volatility Models and
Optimal Stopping �
R�udiger Frey
Department of Mathematics� ETH Z�urich� Switzerlandy
February �� ����
Abstract
In this paper we discuss the superreplication of derivatives in a stochastic volatility
model under the additional assumption that the volatility follows a bounded process�
We characterize the value process of our superhedging strategy by an optimal�stopping
problem in the context of the Black�Scholes model which is similar to the optimal
stopping problem that arises in the pricing of American�type derivatives� Our proof
is based on probabilistic arguments� We study the minimality of these superhedging
strategies and discuss PDE�characterizations of the value function of our superhedging
strategy� We illustrate our approach by examples and simulations�
Key words� Stochastic volatility� Optimal stopping� Incomplete markets� Superreplica�
tion
JEL classi�cation� G��� G��
Mathematics Subject Classi�cation ������� ��A��
� Introduction
The pricing and hedging of derivative securities is nowadays well�understood in the context
of the classical Black�Scholes model of geometric Brownian motion� However� recent
empirical research has produced a lot of statistical evidence that is dicult to reconcile
with the assumption of independent and normally distributed asset returns� Researchers
have therefore attempted to build models for asset price uctuations that are exible
enough to cope with these empirical de�ciencies of the Black�Scholes model� In particular�
a lot of work has been devoted to relaxing the assumption of constant volatility in the
Black�Scholes model and there is a growing literature on stochastic volatility models �SV�
models � see e�g� Ball and Roma ����� or Frey ����� for surveys� In this class of models
the stochastic di�erential equation �SDE that governs the asset price process is driven by
a Brownian motion� but the di�usion coecient of this SDE is modelled as a stochastic
process which is only imperfectly correlated to the Brownian motion driving the asset
price process�
�I would like to thank H� Pham� N� Touzi� N� El Karoui and C� Sin for helpful remarks and interesting
discussions and an anonymous referee for the careful reading of an earlier version of the manuscript�
Financial support from UBS is gratefully acknowledgedypostal address� Department of Mathematics� ETH Zentrum� CH����� Zurich� Switzerland� Email
freymath�ethz�ch
�
SV�models are able to capture the succession of periods with high and low activity
we observe in �nancial markets� However� this increase in realism raises new conceptual
problems for the pricing and the hedging of derivative securities� It is well�known that
SV�models are incomplete� i�e� one cannot replicate the payo� of a typical derivative by
dynamic trading in the underlying risky asset ��the stock� and in some riskless money
market account� This reects a real diculty in the risk management of derivative secu�
rities and should therefore not be considered as a disadvantage of this class of models�
Today �the uncertain nature of forward volatility is recognized as one of the main factors
that drive market�making in options and custom�tailored derivatives�� see Avellaneda and
Paras ����� �
Of course� if there is a liquid market for certain standard derivative securities on the
stock� the use of dynamic trading strategies in the stock and in these securities might
restore market completeness� However� this approach is not always viable� To begin with�
there is not always trade in a sucient number of derivative securities on a particular stock�
Even if there are derivative securities available for trading� running a dynamic hedging
strategy in these securities might prove impossible because of prohibitive transaction costs�
Moreover� this approach requires a precise parametric model for the volatility dynamics of
the underlying asset� As volatility is not directly observable� the determination of a good
model for the volatility dynamics and the estimation of the corresponding parameters
poses dicult problems� Hence there is a considerable risk of model misspeci�cation
that might lead to �bad� hedges� This favours approaches to the risk�management of
derivative securities which require dynamic hedging only in the underlying risky asset
and in the money market account� static positions in liquidly traded derivatives can then
be used in a second step in order to improve the accuracy and reduce the cost of the
hedge� Results from the theory of superhedging imply that even in an incomplete market
it is possible to �stay on the safe side� by using a particular dynamic trading strategy
in the underlying stock and in the money market account� see e�g� Delbaen ����� or
El Karoui and Quenez ����� for results on continuous processes� and Kramkov ����� for
generalisations to a general semimartingale framework� The cost of implementing such
a superhedging strategy is given by the supremum of the expected value of the terminal
payo� over all equivalent local martingale measures for the underlying asset�
Unfortunately the concept of superhedging often leads to prices that are too high
from a practical viewpoint� For instance Frey and Sin ����� and Cvitanic� Pham� and
Touzi ����� show that in a typical SV�model where volatility follows an unbounded
di�usion process the cost of establishing such a superhedge for a European call option is
no smaller than the current price of the underlying stock� hence in this class of models
the cheapest superhedging strategy for a European call option is to buy the underlying
asset� Additional assumptions are therefore called for� if one wants to obtain superhedging
strategies which are at least potentially of some practical interest� In this paper we restrict
ourselves to SV�models where the range of the volatility is bounded� Under this additional
assumption we are able to obtain �nontrivial� superhedging strategies for a large class of
derivatives whose payo� may even be path�dependent� These strategies are universal in
the sense that they depend only on the bounds we impose on the volatility and not on a
particular parametric model for the volatility dynamics� We characterize the value process
of our superhedging strategy by an optimal�stopping problem in the context of the Black�
Scholes model� Roughly speaking our result can be phrased as follows� the value of a
superhedging strategy for a European type derivative under stochastic volatility equals
�
value of a corresponding American type derivative under constant volatility� In particular
one can draw on standard numerical methods for the pricing of American type securities
to implement our approach� The proof is based on probabilistic arguments� Our main
tools are the optional decomposition theorem of El Karoui and Quenez or Kramkov and
the results on time�change for continuous martingales�
In practice it may be impossible to determine �nite bounds on asset price volatility
which hold true with certainty� In those cases we interprete our volatility band as con��
dence interval for the range of the future volatility� By construction the success�set of our
strategy � the set where the terminal value of the hedge portfolio is no smaller than the
the payo� of the derivative � contains all asset price trajectories with volatility lying in
the volatility band� Moreover� our approach is relatively robust� if the actual volatility
exceeds one of the volatility bounds by a small amount the resulting loss will typically
be small� Recently F�ollmer and Leukert ����� have developed a general theory of super�
hedging with a given success probability� In their approach the success set is endogenously
determined� it minimizes the superhedging cost over all strategies with a given success
probability� This yields a very elegant theory� However� by construction the terminal
value of the hedge portfolio is zero on the complement of the success�set� Hence in the
approach of F�ollmer and Leukert the occurrence of an event belonging to the complement
of the success�set may immediately lead to large losses�
It is important to know� if for a given parametric SV�model superhedging strategies
can be constructed which are less expensive than our universal superhedging strategy�
In Section � we study this question for a particular class of SV�models where volatility
follows a one�dimensional di�usion� Most parametric models from the �nancial literature
belong to this class� We show that our universal superhedging strategy is in fact a minimal
superhedging strategy� provided that the bounds on volatility are sharp and that the lower
volatility bound is zero� This generalizes the main result of Frey and Sin ����� � it extends
also certain results of Cvitanic� Pham� and Touzi ����� to path�dependent derivatives�
In most work on superreplication in SV�models with bounded volatility the superhedg�
ing cost is characterized by a terminal value problem involving a parabolic PDE� which is
in general nonlinear� Important examples of this work are El Karoui� Jeanblanc�Picqu�e�
and Shreve ����� � Avellaneda� Levy� and Paras ����� and Lyons ����� � In Section �
we therefore discuss under which conditions the value function of our superhedging strat�
egy can be characterized in terms of some nonlinear parabolic PDE� This gives us also
information on the minimality of our universal superhedging strategy in models where the
lower volatility bound is strictly positive�
In order to illustrate our approach to superhedging we compute in Section � for certain
examples the value function of our strategy� We present simulations for the superrepli�
cation cost of a call spread and compare our results to those of Avellaneda� Levy� and
Paras ����� � We give analytic results on the superhedging cost for a particular barrier
option� namely the down�and�out call option� Finally we present an example that shows
how static positions in traded derivatives can be used for a reduction of the superhedging
cost� an idea which is explored more systematically in Avellaneda and Paras ����� �
�
� Superreplication strategies and optimal stopping
��� The general stochastic volatility model
We consider a frictionless �nancial market with continuous security trading where a risky
asset �the stock and a zero coupon bond with maturity T are traded� In our model the
short rate of interest is deterministic and given by some constant r � � such that B�t� T �
the price of the zero coupon bond at time t� is given by B�t� T � exp��r�T�t � The priceof the stock is modelled as a stochastic process S � �St t�� on some �ltered probability
space ���F � �Ft � P with �Ft satisfying the usual conditions� For the purposes of this
paper it is legitimate to assume that P is already a risk�neutral measure for S� More
precisely� we assume that the dynamics of S are of the following form�
Assumption �� �general stochastic volatility model Consider an �Ft �predictable pro�
cess ��t t�� on ���F � �Ft � P with P ��t � � for all t� � P �R t� �
�sds � � for all t� � ��
The stock price process S solves the SDE
dSt � St��tdWt � rdt ����
for a Brownian motion �Wt t�� on ���F � �Ft � P �
This class of SV�models is very general� In fact� it can be shown that in every arbitrage�
free asset price model where the price process follows continuous trajectories with abso�
lutely continuous quadratic variation the asset price dynamics are of the form ���� � see
for instance Gallus ����� � Obviously Assumption � is satis�ed by most SV�models from
the �nancial literature where volatility is assumed to follow a one�dimensional di�usion�
see Section � for examples�
Fix some maturity date T � By Zt �� er�T�t�St we denote the price of the forward
contract on S with maturity T � The following set of probability measures Q equivalent to
P on ���FT will be important�
Me �� fQ jQ � P and �Zt ��t�T is a Q�local martingaleg �
For further use we also de�ne the process Mt ��R t� �sdWs� M is a continuous local
martingale under all Q � Me with quadratic variation hMit �R t� �
�sds� By It o!s formula
S is given by St � S� exp�rt�Mt � ��hMit �
Remark ���� We will use the following notation�
�i For a process X which is cadlag we put
Xmin���t� �� min
��s�tXs and Xmax
���t� �� max��s�t
Xs �
�ii Let �Gt be a �ltration on some probability space ���F � P and �� and �� be �Gt �stopping times such that �� � �� a�s� We denote by G����� the set of all �nite
�Gt �stopping times � with �� � � � �� a�s�
�iii By �Bt we denote the canonical �ltration on C������ the space of all continuos
functions from ���� to R�
�
��� Construction of superreplication strategies via optimal stopping
We consider the following class of contingent claims�
Assumption �� The payo� H of the contingent claim is of the form
H � f�ZT � Z
min���T �� Z
max���T �
�����
for some function f � R� � R such that the process ft �� f
�Zt� Z
min���t� � Z
max���t�
���t�T
is
bounded below and cadlag�
This class of payo�s comprises all path�independent options� Most common path�
dependent options also satisfy Assumption �� if we assume that the payo� is de�ned as a
function of the forward price Z of the stock� For instance the payo� of barrier options with
barrier condition imposed on Z is of the form ���� � Note that the payo� of a portfolio of
derivatives where each individual contract is of the form ���� is again of this form� This
facilitates the application of our method to portfolios of derivatives�
We now give a formal de�nition of superreplication strategies�
De�nition ���� Consider a contingent claim with maturity date T whose payo� H is
bounded below� A dynamic trading strategy ��� � � ��t� �t ��t�T in stock and bond is a
superreplicating or superhedging strategy for H if
�i The strategy is admissible� i�e� � is predictable� � is adapted� the integralR t� �
�s�
�sS
�sds
is a�s� �nite and the value process Vt � �tSt � �tB�t� T is bounded below�
�ii The terminal value VT of the strategy equals H� Moreover� the cost process associated
with the strategy is nonincreasing� i�e� we have for all � � t � T the representation
Vt � V� �
Z t
��srB�s� T ds�
Z t
��sdSs � Ct ����
for an non�increasing process C � �Ct ��t�T with C� � ��
A superreplicating strategy � �� � whose value process V satis�es for all � � t � T and
for any other superhedging strategy ��� � with value process V the inequality Vt � Vt is
called minimal� The value process V of this strategy� which is uniquely de�ned� is called
the ask�price of the claim H�
Remark ��� Superhedging strategies can be characterized using discounted quantities�
in our case most conveniently in terms of the forward price Zt � StB�t� T � A strategy
��� � with value process V is a superreplicating strategy for H if and only if the discounted
value process "Vt � VtB�t� T � �tZt � �t admits a representation of the form
"Vt � "V� �
Z t
��sdZs � "Ct ����
for a decreasing process � "Ct ��t�T with "C� � �� The cost processes C and "C are related
via d "Ct � exp�r�T � t dCt� The proof� which is an easy application of It o!s product rule�
will be omitted�
In this section we make the following assumption on the asset price dynamics�
�
Assumption � �volatility bounds There are constants �min and �max such that
� � �min � �t � �max �� for all t � � � ����
The numbers �min and �max reect expectations about future volatility� For instance
one could use econometric techniques in order to obtain an estimate for the distribution
of historical volatility and choose �min and �max as some lower respectively upper quantile
of this distribution� in that case the interval ��min� �max� can be interpreted as con�dence
interval for the future volatility� If there is a liquid market for derivative instruments on
S� one could alternatively obtain �min and �max from extreme past values of the implied
volatilities of these contracts� In either case the volatility band should be wide enough to
ensure that current implied volatilities of liquidly traded derivatives are contained in the
band� Otherwise the use of static positions in these instruments as additional hedging
tool might lead to inconsistencies� see Section ����
Consider a claim H satisfying Assumption �� and two numbers � � � � � � � with
� � �� Denote by Rz the law of the solution of the SDE dUt � UtdWt with initial
value U� � z and recall the de�nition of the set of stopping times B���T�t�� ���T�t� from
Remark ���� De�ne the function V Am � ��� T ��R� � R as value function of the following
optimal�stopping problem
V Am�t� z�m�m��� � ��
supnERz
hf�U�� m Umin
������ m Umax�����
i� � B���T�t�� ���T�t�
o� ����
V Am has an obvious interpretation as arbitrage price of an American type derivative with
partial exercice feature in a standard Black�Scholes model with volatility equal to one
and interest rate equal to zero� Wider �volatility bounds� � and � correspond to a larger
time window for the exercice of this American�type security and hence to a larger value
of V Am�
The following theorem is the main result of this section�
Theorem ��� Suppose that Assumptions � and � hold for S� that H satis�es Assumption
� and that the function V Am�t� x�m�m��min� �max is �nite for all �t� x�m�m � ��� T ��R��
Then the process V � � �V �t ��t�T de�ned by
V �t �� e�r�T�t�V Am
�t� Zt� Z
min���t� � Z
max���t� ��min� �max
�
is the value process of a superreplicating strategy for H�
Comments�
Consider a payo� of the form H � f�ST for some convex function f � By Jensens
inequality we get for any stopping time � B��min�T�t����max�T�t�
ERz �f�U� � � ER
z
�f�ERz �U��max�T�t�
jB� ��� � ER
z �f�U��max�T�t� � �
It follows that the sup in ���� is attained by taking � ��max�T � t � Hence V �t equals
the price of a derivative with payo� f�ST in a Black�Scholes model with volatility �max�
see El Karoui� Jeanblanc�Picqu�e� and Shreve ����� �
If H is of the form H � f�ST the value function of the optimal�stopping problem ����
can be computed using the standard binomial model of Cox� Ross� and Rubinstein ����� �
�
If one is dealing with path�dependent payo�s some algorithm for the pricing of American
path�dependent options such as the forward shooting grid method of Barraquand and
Pudet ����� can be used�
Note that the superreplication cost V � is subadditive� i�e� the superhedging cost
corresponding to a portfolio of two payo�s is no larger than the sum of the superreplication
costs of the two individual payo�s� In order to keep the superreplication cost low the
method should therefore be applied to large portfolios rather than to individual derivatives�
Our approach can easily be adapted to accommodate portfolios of claims with di�erent
maturity dates� see also Avellaneda� Levy� and Paras ����� � Consider the case of two
claims H� and H� with � for notational simplicity path�independent � payo�s f��ST�
and f��ST� and maturity dates T� � T�� By Theorem ���
V ����ST� �� exp��r�T� � T� VAm�T�� exp�r�T� � T� ST�
is the value at time T� of the superreplicating strategy for H�� De�ne a new claim
H with maturity date T� and payo� given by H � f��ST� � V ����ST� � Obviously� a
superreplicating strategy for H� which can be computed using Theorem ���� induces a
superreplicating strategy for the portfolio consisting of the claims H� and H��
��� Proof of Theorem ���
As a �rst step we recall some well�known recent results on the existence of minimal super�
replicating strategies and in particular the optional decomposition theorems as obtained
by El Karoui and Quenez and by Kramkov� the following result is a version of Theorem
��� in Kramkov ����� �
Theorem ���� �optional decomposition Let Z be a locally bounded process on some
stochastic basis ���F � �Ft � P such that the set Me of equivalent local martingale mea�
sures is non�empty� Then a positive process � "Vt t�� is a supermartingale for all Q � Me
if and only if there is a predictable� Zintegrable process ��t t�� and a decreasing� adapted
process � "Ct t�� with "C� � � such that
"Vt � "V� �
Z t
��sdZs � "Ct � ����
This theorem obviously extends to processes which are only bounded below� Together
with the following result of El Karoui and Quenez ����� � Theorem ��� ensures the ex�
istence of minimal superhedging strategies and gives moreover a characterization of the
ask�price in terms of the set Me�
Proposition ���� Let H be a contingent claim with maturity T and payo� which is
bounded below� Suppose that supfEQ�H� � Q �Meg � � � Then there is a RCLL pro�
cess H � �Ht t�� such that Ht � ess supfEQ�HjFt� � Q � Meg� Moreover� the process H
is a Q�supermartingale for all Q �Me�
It is easily seen from ���� that whenever ��� � is a superreplicating strategy for H
we must have "Vt � EQ�HjFt� for all Q � Me� as the stochastic integralR t� �sdZs is a Q�
supermartingale� Hence "Vt � Ht� and we have the following result� which is well�known
in the theory of superreplication�
Corollary �� � Let H be a contingent claim satisfying the hypothesis of Proposition ���
Then a minimal superhedging strategy exists� the ask�price is given by �e�r�T�t�Ht ��t�T �
�
We next explain the key idea behind the characterization of superhedging strategies
given in Theorem ���� In view of Corollary ��� we want to give an estimate for H�
Consider for notational simplicity some path�independent payo� H � f�ST � Then we
have for every Q �Me�
EQ�f�ST � � EQ�f�ZT � � EQ�f�Z� exp�MT � ��hMiT � �By changing the volatility for t � T if necessary we may assume that limt��hMit � �P�a�s� De�ne the increasing process At via
At � A�t �� inffs � � � hMis � tg � ����
Note that under our assumptions on the volatility the mapping t� hMit is P�a�s� a bijec�
tion from ���� onto itself with inverse mapping given by A� Now Levy!s characterization
of Brownian motion implies that the process Bt �� MAt is a Brownian motion relative
to the new �ltration �Gt � �FAt and Mt � BhMit � see e�g� Chapter ��� of Karatzas
and Shreve ����� � Moreover hMiT is a �Gt �stopping time which takes its values in the
interval ���minT� ��maxT � by Assumption �� Hence we get
EQ�f�ST � � EQ�f�Z� exp�BhMiT � ��hMiT � ����
� sup�EQ�f�Z� exp�B� � �� �� � G��minT��
�maxT
�� �����
We will show in the proof of Proposition ���� below that the strong Markov property of
the geometric Brownian motion Ut �� Z� exp�Bt���t implies that the value function of
the optimal�stopping problem ����� is independent of the particular �ltered probability
space on which U is de�ned and equal to V Am��� Z� � Hence the ask price of H is no
larger than V �� �
Remark ���� Obviously� these estimates are valid in the case where �max �� and where
the payo� under consideration is path�dependent but satis�es Assumption �� Hence for
every such claim and every � � t � T the ask price is no larger than
V �t � e�r�T�t�V Am
�t� Zt� Z
min���t� � Z
max���t� ��min��
��
Remark ���� The above argument cannot be extended to the case of options on multiple
assets S�� � � � � Sd with payo� f�S�T � � � � � S
dT � Assume that the dynamics of Si are of the
form dSit � SitdMit for continuous local martingales M i� i � �� � � � � d with absolutely
continuous quadratic variation hM iit� To illustrate� what �goes wrong� we consider the
particular case where
hM i�M jit � � for all � � i �� j � d and for all t � � �
As previously we introduce Ait �� inf
�s � �� hM iis � t
�and de�ne new processes Bi�
� � i � d by Bit �� MAi
t� The F� B� Knight Theorem �see for instance Karatzas and
Shreve ����� Theorem ������ ensures that B � �B�� � � � � Bd is a d�dimensional stan�
dard Brownian motion w�r�t� its own �ltration �GBt However� B is in general not a
d�dimensional Brownian motion w�r�t� the �ltration �Gt de�ned by
Gt � FA�t � � � � FAd
t�
On the other hand� while hM iit is for every � � i � d and every t a �Gt �stopping time�
hM iit is not necessarily a stopping time for the �ltration �GBt � Hence we cannot hope to
�nd a representation for EQ�f�S�T � � � � � S
dT � of the form ���� in this case�
�
In view of Corollary ���� Theorem ��� would follow from the previous estimates if
under Assumptions � and � V �t was actually equal to e�r�T�t�Ht for all t� As shown in
Sections � and � below� this equality holds true for a large class of general SV�models but
is wrong in general� However� Theorem ��� follows from Theorem ��� and the following
proposition�
Proposition ����� The process V Am�t� Zt� Z
min���t� � Z
max���t�
���t�T
is a Q�supermartingale for
all Q �Me�
Proof of Proposition ����� De�ne for every � � t � T positive random variables
�min�t and �max�t via
�min�t �� A���min�T � t � hMit and �max�t �� A���max�T � t � hMit � �����
Lemma ���� below shows that �min�t and �max�t are �Ft �stopping times� Moreover�
t � �min�t � �max�t P�a�s� Recall the de�nition of the process ft �� f�t� Zt� Z
min���t� � Z
max���t�
�in Assumption � and the de�nition of the set of �Ft �stopping times F�min�t���max�t� in
Remark ����ii � Fix some Q � Me and de�ne a process JQt via the following optimal
stopping problem
JQt �� ess supfEQ�f� Ft� � � � F�min�t���max�t�g � �����
The proof now consists of two steps�
Step �� JQt is a Q�supermartingale�
To prove the supermartingale property note �rst that the set of stopping times F�min�t���max�t�
is shrinking as t increases� We get that
�
�t�min�t � A����min�T � t � hMit ����min � ��t � � �
�
�t�max�t � A����max�T � t � hMit ����max � ��t � ��
The inequalities follow as A��x � � for all x � � and as ����min � ��t � � and ����max �
��t � � by Assumption �� Now let t � s� We get that
EQ�JQt Fs� � EQ
�ess supfEQ�f�
Ft� � � � F�min�t���max�t�g Fs�
�i�� ess supfEQ�f�
Fs� � � � F�min�t���max�t�g�ii�
� ess supfEQ�f� Fs� � � � F�min�s���max�s�g
� JQs �
Here the equality �i follows from Proposition A�� in Appendix A��� inequality �ii follows
as F�min�t���max�t� � F�min�s���max�s� for t � s�
Step �� JQt is independent of Q and given by V Am�t� Zt� Z
min���t� � Z
max���t�
��
For notational simplicity we treat only the case t � �� We want to write JQ� in a di�erent
way using the time change introduced in the beginning of the proof� De�ne the process
U via Ut � Z� exp�Bt � ��t � where Bt �MA�t� is Q�Brownian motion� We have
Zt � UhMit � Zmin���t� � Umin
���hMit�and Zmax
���t� � Umax���hMit�
� �����
�
By ����� we get that
EQ�f� � � EQhf�UhMi� � U
min���hMi� �
� Umax���hMi� �
�i
for every �Ft �stopping time � � The following Lemma� whose proof is given in the Ap�
pendix A��� shows that the mapping � � hMi� is a bijection from F�min�����max��� onto
G��minT� ��maxT
�
Lemma ����� Let � be an �Ft �stopping time� Then �� �� hMi� is a �Gt �stoppingtime� Conversely� if is a �Gt �stopping time� �� �� A� is an �Ft �stopping time�
Using this Lemma and Proposition A�� we can write JQ� in a di�erent way�
JQ� � supnEQ
hf�U� � U
min������ U
max�����
i� � G��minT� �
�maxT
o� EQ
hess sup
nEQ
hf�U� � U
min������ U
max�����
F��minT
i� � G��minT� �
�maxT
oi� �����
Now U is a Q�geometric Brownian motion with zero drift� initial value U� � Z� and volatil�
ity equal to one� in particular U is Markovian� Hence the process U t ��Ut� U
min���t� � U
max���t�
�is a R��valued Markov process� Thisd implies that the Snell�envelope of the process
ft � f�U t with respect to the �ltration �Gt coincides with the Snell envelope of f�U t
with respect to the �ltration generated by U respectively by U � see Lemma ��� of Muli�
nacci and Pratelli ����� or Proposition ��� of Lamberton and Pag#es ����� � Hence
ess supnEQ
hf�U�� U
min������ U
max�����
F��minT
i� � G��minT� �
�maxT
o�
� ess supnERZ�
hf�U� � U
min������ U
max�����
B��minT
i� � B��minT� �
�maxT
o�
� V Am��� U��minT
� Umin�����minT �
� Umax�����minT �
� �� ���max � ��min ���
��
Plugging this into ����� yields
JQ� � EQhV Am
��� U��minT
� Umin�����minT �
� Umax�����minT �
� �� ���max � ��min ���
�i� ER
Z�
hV Am
��� U��minT
� Umin�����minT �
� Umax�����minT �
� �� ���max � ��min ���
�i� V Am��� Z���min� �max �
where the last equality follows from the de�nition of V Am� if we condition on B��minT�
apply Proposition A�� and the Markov�property of U �
� Minimality of our superhedging strategies
In this section we study under which conditions the superhedging strategy constructed in
Theorem ��� is actually a minimal superhedging strategy� To analyze this question we have
to introduce additional assumptions on the probabilistic structure of the volatility process�
We are particularly interested in the case where the volatility follows a one�dimensional
di�usion�
��
Assumption � We assume that S satis�es the equations
dSt � St�jvtj���dW ���t � rdt � ����
dvt � a�vt dt� ���vt dW���t � ���vt dW
���t � ����
for Wt � �W���t �W
���t a standard two�dimensional Wiener process on ���F � �Ft � P � We
assume that the coe�cients are such that the vector SDE ����� ���� has a non�exploding
and strictly positive solution�
This class of volatility models contains the models considered by Wiggins ����� � Hull
and White ����� or Heston ����� as special cases� Note that we allow for nonzero ��and hence for nonzero instantaneous correlation between volatility innovations and asset
returns�
Theorem ��� Suppose that S is given by a SV�model satisfying Assumption �� Assume
moreover that there is some � � �max � � such that
�i The real functions a� ��� �� are locally Lipschitz on ��� ��max � b�x ��p����x � ����x
belongs to C������max��
�ii ���v � � for all v � ��� ��max �
�iii � � �t ��pvt � �max� the last inequality being strict for �max ���
Then for every claim H satisfying Assumption � the ask price equals
V �t � e�r�T�t�V Am
�t� Zt� Z
min���t� � Z
max���t� � �� �max
�for all � � t � T � ����
Comments�
Hypothesis �ii ensures that volatility innovations and asset returns are not perfectly
correlated which in turn implies that the market is incomplete� Moreover� this hypothesis
ensures that for all t � � and all � � K� � K� � �max we have that P ��t � K�� � �
and P ��t � K�� � �� i�e� the open interval ��� �max is contained in the range of �t for
all t � �� As we assumed existence of a strictly positive solution to ���� � hypothesis �i
implies pathwise uniqueness for this SDE� see Frey and Sin ����� for details�
Consider models with unbounded volatility� i�e� �max ��� Applying Theorem ��� to
ordinary call options we get that in a large class of SV�models where the volatility follows
a one�dimensional di�usion the ask price of a call option is equal to S�� the current price
of the stock� This is the main result of Frey and Sin ����� �
Cvitanic� Pham� and Touzi ����� have previously obtained the characterization of
the ask price by the optimal�stopping problem ���� for a large class of di�usion models
with unbounded volatility and path�independent payo�s� Their approach is based on
the characterization of the ask price as viscosity supersolution to the Bellman equation
corresponding to the in�nitesimal generator of the process S� Using that characterization
they conclude that the ask price is given by the smallest concave majorant f� of f � In
Lemma ��� of their paper it is moreover shown that the solution to the optimal�stopping
problem ���� is equal to f��
As shown below� Theorem ��� follows from combining results from Frey and Sin �����
with the following Proposition which applies to general Markovian SV�models��
�The author is grateful for interesting discussions with N� Touzi and H� Pham� which were very helpful
in obtaining Proposition ����
��
Proposition ��� Consider a model where S is given by a stochastic process satisfying
Assumption �� Suppose that there is some � � � �� such that the following holds�
�� The process Xt �� �St� �t is a two�dimensional strong Markov family de�ned for all
initial values X� � �S�� �� � R � ��� � � The corresponding family of measures will be
denoted by �Px � x � R � ��� � �
�� For every � � � and every x � R���� � there is a sequence of strictly positive density
martingales G��n � �G��nt ��t�T with G��n
� � � such that
�i G��n is adapted to the �ltration generated by X�
�ii The process Zt � er�T�t�St� � � t � T is a local martingale under the probability
measures Q��nx de�ned by dQ��n
x dPx � G��nT �
�iii limn��Q��nx �hMiT � ���T � � � � ��
�� For every compact set K �� R � ��� � and every � � � there is a sequence G��n of
strictly positive density martingales G��n � �G��nt ��t�T with G��n
� � � such that
�i G��n is adapted to the �ltration generated by X�
�ii Z is a local martingale under the measures Q��nx de�ned by dQ��n
x dPx � G��nT �
�iii limn�� infx�K Q��nx �hMiT � � � � ��
Then for every claim H satisfying Assumption � the ask price is no smaller than
e�r�T�t�V Am�t� Zt� Z
min���t� � Z
max���t� � �� �
��
Proof of Proposition ����
While the following proof is rather technical� the underlying idea is simple� Recall the
�ltration �Gt introduced in the proof of Proposition ����� We want to show that for every
� G���� and every � � � there is a sequence Qn �Me such that Qn�hMiT � �� ����� �
as n � �� Together with the right�continuity of our payo�s this implies the result� To
construct such a sequence of local martingale measures we �rst choose a sequence of
measures Q��n �Me which put most of the mass on trajectories with �high� volatility� As
soon as hMit � we �drive the volatility down� using another sequence Q��n �Me�
The crucial step in the proof is the following Lemma�
Lemma �� Suppose that the hypothesises of Proposition ��� hold� Then for every �G���� and for every � � � � there is some Q �Me such that
Q�hMiT � �� � ��
�� �� � ����
The proof of Lemma ��� is given in Appendix A���
We now show that Proposition ��� follows from Lemma ���� By the Markov property
of X it is enough to consider the case t � �� As in the proof of Theorem ��� we de�ne the
process Ut �� ZA�t�� We get
H � f�UhMiT � U
min���hMiT �
� Umax���hMiT �
��
Let " be a stopping time for �Bt � the canonical �ltration on C������ Then �� " � Uis a stopping time for �GUt � the sub��ltration of �Gt generated by the process U � and
��
hence for �Gt � Moreover� we have for every Q � Me that EQ �f�U� � � ERZ�
�U� �� as U is
Q�geometric Brownian motion�
Choose some " � B����T and the corresponding stopping time �� " � U from G����T �By Lemma ��� there exists a sequence of local martingale measures Qn � Me such that
Qn �hMiT � �� � ��� � � � �n� By Assumption � the process t � f�Zt� Z
min���t� � Z
max���t�
�is right continuous such that
limn��
QnhH � f
�U� � U
min������ U
max�����
� � �i� � ����
for all � � �� Consider �rst the case of bounded f � It follows from ���� that
lim infn��
EQn
�H� � lim infn��
EQnhf�U� � U
min������ U
max�����
�i� ER
Z�
hf�U� � U
min������ U
max�����
�i�
As " � B���� was arbitrary we get that lim infn��EQn�H�� � V Am��� Z� � In the case
of unbounded but positive f we consider for n � N the claim Hn corresponding to the
payo� fn �� f n� Applying the result for bounded f we get that H is greater than
V Amn ��� Z� � the value function of the optimal�stopping problem ���� with f replaced by
fn� Monotone integration now implies that H � V Am��� Z� �
Proof of Theorem ����
We have to show that Conditions � and � of Proposition ��� are satis�ed under the
assumptions of Theorem ���� Our argument is based on results obtained by Frey and Sin
����� for models with unbounded volatility� If �max � � we transform our problem to
the case �max � � using some smooth and strictly increasing function � that maps the
interval ��� ��max onto ���� � By It o!s formula yt �� ��vt solves the SDE
dyt � "a�yt dt� "���yt dW���t � "���yt dW
���t �
where the coecients "a� "��� "�� and "b ��p
"��� � "��� satisfy hypothesis �i and �ii of Theo�
rem ��� on ���� � As in Frey and Sin ����� we consider for n � N measures Q��n and
Q��n �Me with densities given by
dQ��n
dP� exp
nW
���t � �
�n�T
�and
dQ��n
dP� exp
�nW ���
t � �
�n�T
��
As �t �pvt and ���vt are strictly positive� the �ltration generated by the two�dimensionalX ��
�S� v coincides with the �ltration generated by �W ����W ��� � see e�g� Harrison and Kreps
����� � Hence our density martingales are adapted to the �ltration generated by X� By
Girsanov!s theorem �yt t�� is under Qi�n� i � �� �� a solution to the SDE
dynt � "a�ynt � ��� in"���ynt dt�"b�ynt dBi�nt � y� � ��v� � ����
for a new Qi�n�Brownian motion Bi�n� The next two results are Lemma ��� and Lemma ���
of Frey and Sin ����� �
Lemma �� Assume that the SDE ��� has a global and strictly positive solution for
any n � N� any initial value y � R and any i � f�� �g� and denote by Ri�ny the law of
��� with initial value y� Then
�i For every L � �� T � �� y � � and � � there exists N� � N such that
R��ny �yt � L for some � � t � T � � �� for all n � N��
��
�ii For every L � �� T � �� y � � and � � there exists N� � N such that
R��ny
�yt � L�� for some � � t � T
�� �� for all n � N��
Lemma ��� Assume again that for n � N� y � R and for i � �� � the SDE ��� has a
global solution which is strictly positive� Then the following holds�
�i For every L � �� T � �� and � � there exists N� � N such that for y� � �L
R��n�L �yt � L for all � � t � T � � �� for all n � N��
�ii For every L � �� T � � and � � there exist N� � N such that for y� � L�
R��nL�� �yt � L for all � � t � T � � �� for all n � N��
To verify that Conditions ���iii and ���iii of Proposition ��� are implied by Lemmas
��� and ��� one now uses exactly the same arguments as in the proof of Frey and Sin
����� � Theorem ����
� PDE�Characterisation of the value function VAm
��� Previous results
In most of the previous work on superreplication in stochastic volatility models the value
process of superhedging strategies is characterized by a terminal value problem involving
some � often nonlinear � parabolic PDE� Important examples of this work are the
independent papers Avellaneda� Levy� and Paras ����� � Lyons ����� and El Karoui�
Jeanblanc�Picqu�e� and Shreve ����� � These papers consider mainly path�independent
derivatives� Therefore we will concentrate on payo�s of the form H � f�ST for some
continuous function f � Moreover� we start immediately with discounted quantities�
The main result of Avellaneda� Levy and Paras and Lyons can be stated as follows�
Proposition ��� Suppose that S satis�es Assumptions � and � and that the terminal
value problem
hAVt ��
�x�
����min
�hAVxx
��� ��max
�hAVxx
��� � � hAV�T� x � f�x ����
has a solution in C������� t �R � Then hAV�t� St is the value at time t of a superhedging
strategy for H�
For convenience we recall the simple proof� We get from It o!s formula
f�ST � hAV�T� ST � hAV��� S� �
Z T
�hAVx �t� St dSt
�
Z T
��hAVt �
�
�S�t �
�t h
AVxx �t� St � z �
� � by ����
dt �
such that the strategy with value process hAV�t� St and stockholdings hAVx �t� St has a
representation of the form ���� �
��
Remark ��� Lyons ����� has developed an extension of this result to markets with
more than one risky asset� Cvitanic� Pham� and Touzi ����� prove that the terminal
value problem ���� admits a classical solution if �min � � and if the payo� is suciently
smooth� Moreover� they show that in a large class of SV�models of the form ���� � ����
with ���v � � for all v � ���min� ��max the ask price of a claim with payo� f�ST is no
smaller than hAV�t� St � provided of course that a solution to ���� exists�
Remark �� Note that the above argument also works for functions hAV � C������� t �R � if the space derivative hAVx �t� � is moreover absolutely continuous in x for every t� For
an extension of It o!s formula to such situations see e�g� �Krylov ����� Theorem ������ �
We now discuss the relation between V Am and the nonlinear PDE ���� � For this we
have to distinguish the cases �min � � and �min � ��
��� The case �min � �
In this case the value function V Am of our superhedging strategy will typically not belong
to C������� T �R � as the second derivative V Amxx is usually discontinuous at the optimal
stopping boundary of the optimal�stopping problem de�ning V Am� see also Section ���
below� We therefore contend ourselves with a local result�
Proposition �� Assume that �min � ��
�i If the value function V Am�t� St� �� �max de�ned in ��� is of class C��� in some open
set B �� ���� T � R � V Am solves the following version of the PDE �����
V Amt �t� x �
�
�x���max�V
Amxx �� x � � � for all �t� x � B � ����
�ii Suppose that there is a solution hAV of the terminal value problem ����� which
belongs to C������� t �R and whose space derivative hAVx �t� � is moreover absolutelycontinuous� Then V Am � hAV�
Proof� We start with �i � From the characterization of solutions to the optimal�stopping
problem ���� via variational inequalities we get for all �t� x � B
V Amt �t� x �
�
�x���maxV
Amxx �t� x � � ����
V Am�t� x � f�x and V Am�T� x � f�x � ����
where at least one of the two inequalities must holds with equality� For a proof see e�g�
Jaillet� Lamberton� and Lapeyre ����� or Myeni ����� � Moreover� V Am is decreasing in
t� i�e� we have V Amt � � in B�t�� x� � Choose some �t�� x� � B� Now we distinguish two
cases�
�a V Amxx �t�� x� � �� We shall show that this implies V Am�t�� x� � f�x� � hence equality
must hold in ���� which shows that ���� holds in this case� Assume to the contrary
that V Am�t�� x� � f�x� � in that case we must have V Amt �t�� x� � �� as V Am
t �t�� x� � �
would yield a contradiction to ���� � However� together with V Amxx �t�� x� � � this implies
that V Amt �t�� x� � V Am
xx �t�� x� � � which contradicts ���� �
�b V Amxx �t�� x� � �� We show that in that case V Am
t �t�� x� � �� which implies the
result� Assume to the contrary that V Amt �t�� x� � �� Hence strict inequality holds in
���� such that ���� must hold with equality� However� together with V Amt �t�� x� � �
this contradicts ���� which proves that we must have V Amt �t�� x� � ��
��
Let us now turn to �ii � As shown before hAV induces a superhedging strategy in all
SV�models satisfying Assumptions � and �� hence in all models satisfying the hypothesis
of Theorem ���� As V Am is minimal in these models we have the inequality hAV � V Am�
The converse inequality is proved in �Cvitanic� Pham� and Touzi ����� Remark ��� �
��� The case �min � �
To study the relation between V Am and solutions to the nonlinear PDE ���� we de�ne
the function u� R � R � R � R by
u�t�� t�� x �� sup�ERx �f�U� � � � Bt��t�t�
�� ����
where �Bt denotes again the canonical �ltration on C������ The function V Am is related
to u via
V Am�t� x��min� �max � u���min�T � t � ���max � ��min �T � t � x � ����
Now� using Proposition A�� we may express u as follows�
u�t�� t�� x � ERx
�ess sup
�ERx �f�U� jFt� � � � Bt��t�t�
��� ER
x �h�t�� Ut� � �
where h is de�ned via the following standard optimal�stopping problem
h�t� x � sup�ERx �f�U� � � � B��t
�� ����
We make the following regularity assumption on h�
Assumption �� The function h de�ned in ���� is continuous on ����� � R and
is of class C�������� � R � Moreover� for every t there is a �nite number of points
x�� � � � � xn�t�� such that for all x � R � fx�� � � � � xn�t�g there is an open environment
B�t� x � ����� � R where h is twice continuously di�erentiable in x� Moreover the
functions ht� xhx� hxx and x�hxx are uniformly bounded on ���� � R�
Remark ��� These regularity assumptions typically hold for the value function of the
optimal�stopping problem ���� � provided that the terminal value f is suciently smooth�
see for instance the examples in Section ����
Proposition ��� Suppose that �min � �� Under Assumption � we have the following
�i u belongs to C���������� � ���� � R �
�ii We have for all �t�� t�� x � ���� � ���� � R
ut� � �t�� t�� x ��
�x�uxx�t�� t�� x � ����
ut��t�� t�� x � ERx
�x��U�
t�hxx�t�� UT� � � �
�x��uxx�t�� t�� x �
� ����
Equality in ���� holds if and only if hxx�t�� � is either everywhere nonnegative or
everywhere nonpositive�
�iii V Am�t� x��min� �max satis�es the following di�erential inequality�
V Amt �
�
�x��� ��min�V
Amxx �� � ��max�V
Amxx �
� � � � �����
Equality holds if and only if equality holds in ����� in particular for f convex on
R or f concave on R� Moreover� we have V Am�t� x � hAV�t� x � equality holds if
and only if ����� holds with equality�
��
The most important result here is �iii � This result implies that in a model with
strictly positive lower volatility bound the ask price of a derivative whose payo� is neither
everywhere convex nor everywhere concave may be smaller than V Am� However� as shown
in Section ���� numerical values of V Am and hAV are typically close to each other�
Proof� De�ne for t� �xed the function "u�t� x �� u�t� t�� x � It follows immediately from
the Feynman�Kac formula that "u solves the initial value problem
"u�t� x ��
�x�"uxx�t� x � "u��� � � h�t�� � �
Hence u is C� in t�� C� in x and it satis�es ���� � By Proposition ��� and Assumption �
the function h solves for almost all x the PDE ht ���x
��hxx�� Again by Assumption �
we may exchange di�erentiation and expectation yielding
�u
�t��t�� t�� x � ER
x
��h
�t�t�� Ut�
�
��
�ERx
�U�t� �hxx�t�� Ut� �
�
� �
�
hERx
�U�t�hxx�t�� Ut�
� i� �����
where the last estimate follows from Jensen!s inequality� Recall that U is Rx�geometric
Brownian motion with initial value U� � x� hence Ut� is Rx�a�s� equal to x exp�Wt � �� t
such that ��xUt� � Ut�x� We use this information to compute the derivatives of h�t�� Ut�
with respect to x and get
��
�x�h�t�� Ut� �
�
�x
hx�t�� Ut�
Ut�x
��U�t�
x�hxx�t�� Ut� �
As the last expression is bounded by Assumption � we may exchange expectation and
di�erentiation and get
��
�x�u�t�� t�� x � x��ER
x
�U�t�hxx�t�� Ut�
��
Combining this with ����� we get ut��t�� t�� x � ��x
��uxx�t�� t�� x �� i�e� ���� � Obviously
equality holds if and only if equality holds in ����� � which proves �ii � Let us now turn
to �iii � By ���� we get that
V Amt � ���minut� � ���max � ��min ut�
�a�
� ���min
�
�x�uxx � ���max � ��min
�
�x��uxx�
�b�� ��
�x�����min�V
Amxx �� � ��max�V
Amxx � �
which is ����� � Here �a follows from statement �ii and �b from the relation uxx � V Amxx �
The inequality V Am�t� x � hAV�t� x follows now from the maximum principle for viscosity
solutions of nonlinear parabolic PDE!s� see Remark ��� of Cvitanic� Pham� and Touzi
����� �
��
� Examples and simulations
��� Path�independent derivatives
In this section we consider path�independent derivatives whith payo� given by some func�
tion f�ST � As in Section � we distinguish between the cases �min � � and �min � ��
i The case �min � �� In the case of unbounded volatility� i�e� �max � �� V Am is
given by the smallest concave majorant f� of f � see the comments following Theorem ����
Cvitanic� Pham� and Touzi ����� give the following description of f� as ane envelope
of f �
f��x � inf fc � � � �$ � R such that c�$�z � x � f�z for all z � �g � ����
Let us now consider a call�spread with strike prices K� � K� as more speci�c example�
the payo� of this derivative is given by f�x �� �x � K�� � �x � K��
� This payo� is
interesting in our context as it is neither everywhere convex nore everywhere concave�
Hence the superreplication price is not simply the Black�Scholes price corresponding to
one of the volatility bounds� Using the description ���� it is immediately seen that for
�max �� the superreplicating cost is given by
f��x ��
� K��K�K�
x � � � x � K�
K� �K� � x � K�� ����
For �max � � we have to use numerical techniques to obtain values for V Am� Figure �
shows V Am� the superreplication cost for a standard call�spread with K� � ��� K� � ����
time to maturity equal to � month and volatility bounds given by �min � � and �max � ����
Observe that V Am�t� x � K� �K� whenever x � K�� as for x � K� immediate exercice
is the optimal strategy in the stopping problem de�ning V Am�
Remark ���� Note that the left limit limx�K��V Amx �t� x must be larger than �K� �
K� K�� as VAm � f�� the superreplicating cost for �max � �� V Am
x is therefore dis�
continuous in x � K�� hence in particular not absolutely continuous with respect to x�
compare also Figure �� By Proposition ��� �ii the terminal value problem ���� does there�
fore not admit a classical solution�This shows that at least for �min � � the PDE�approach
to superhedging is not always as straightforward as it seems at �rst sight�
In Figure � we have graphed the superhedging cost for a �call�spread� with smooth
terminal payo� f �� time to maturity equal to � month and volatility bounds given by
�min � � and �max � ���� together with the terminal payo� f � Recall the de�nition of the
stopping boundary B� for the stopping problem de�ning V Am� Here B� is given by
B� � f�t� b��t � b��t � inffx � �� V Am�t� x � f�x g �
We see that in the example with smooth terminal payo� we have �smooth �t�� i�e� the
space derivative V Amx is continuous at b�� However� the second derivative V Am
xx is discon�
tinuous at b�� On the one hand we have
limx�b��t��
V Am�t� x � f��
�b��t � � �
�f is given by the Black�Scholes price of a standard call�spread with K� � ��� K� � ��� time to
maturity one week and volatility ����
��
On the other hand it follows from the characterization of V Am via variational inequalities
that V Amxx �t� x � � whenever V �t� x � f�x � see also the proof of Proposition ���� The
regularity properties of this example� which are typical for optimal�stopping problems
with suciently smooth payo�� motivate some of the hypothesises in Assumption ��
�
�
�
�
�
��
��
�� � �� � �� �� ��� ��� ��� ���
V ������ S
S
V �
Figure �� Superreplication cost for a standard call�spread with K� ��� K� ���� time to
maturity � month and volatility bounds �min � and �max ����
ii The case �min � �� We know from Proposition ��� �iii that V Am is typically not
equal to the ask�price of a path�independent derivative whenever the terminal payo� is
of mixed convexity� To get a feeling for the numerical size of the di�erence between V Am
and the discounted ask�price� which is given by the solution hAV to the terminal value
problem ���� � we computed V Am for the standard call�spread considered above� In Table
� we present for di�erent values of S� our solution V Am together with values for hAV
taken from Avellaneda� Levy� and Paras ����� � assuming that the volatility is bounded
by �min � ��� and �max � ���� We see that the di�erence between the two functions is
relatively small�
S� �� �� �� �� ��
V Am ���� ���� ���� ���� ����
hAV ���� ���� ���� ���� ����
V Am � hAV ���� ���� ���� ���� ����
Table �� Superreplication price for a standard call�spread with K� ��� K� ���� time to ma�
turity � month� and volatility bounds �min ��� and �max ���� V Am gives the superreplication
cost according to our approach� hAV is the solution to the terminal value problem ������
��� Barrier options
We now consider a particular barrier option namely a down�and�out call on the forward
price with strike price K and barrier H as example of a derivative with path�dependent
payo�� In the notation introduced in Section ��� its payo� is given by
f�ZT � Z
min���T �
��� �ZT �K��n
Zmin���T �
�Ho �
��
�
�
�
�
�
��
��
�� � �� � �� �� ��� ��� ��� ���
V ������ S
S
V �
f
Figure �� Superreplication cost for a �smooth call�spread� with terminal payo� f�x�� time to
maturity � month and volatility bounds �min � and �max ����
For this particular payo� we may give an analytic expression for the superhedging strategy
V Am� For our analysis we have to distinguish if H � K or if H � K�
i The case H � K� It is well�known that in this case a rational investor will never
exercise an American down�and�out call before maturity� see e�g� Reimer and Sandmann
����� for the corresponding portfolio argument� Hence our superhedging cost V Am equals
the price of the down�and�out call in a Black�Scholes model with constant volatility equal
to the upper volatility bound �max and zero interest rate� independently of �min� This
price is well known� see e�g� Reimer and Sandmann ����� or Chapter � of Musiela and
Rutkowski ����� � By Theorem ��� this is the ask�price for the down�and out call in a
large class of SV�models with volatility range ��� �max�� As the optimum in the stopping
problem for V Am is attained at a deterministic stopping time� the proof of Theorem ���
shows that V Am is the ask price for the barrier option even if �min � ��
Le us now consider the case �max ��� Inspection of the formula for the Black�Scholes
price of our barrier call shows that in that case V Am is given by
V Am�t� Zt� Zmin���t� � �n
Zmin���t�
�Ho�Zt �H �
By Theorem ��� this is the ask�price of the down�and�out call in most of the standard
SV�models with unbounded volatility � The corresponding hedging strategy is a buy and
hold strategy� At t � � we buy one share of the stock and sell H zero�coupon bonds
with maturity T � If the barrier is hit the value of our position is zero and we dissolve the
portfolio immediately� otherwise we hold our position until maturity�
ii The case H � K� An elementary argument shows that for �min � � the function
V Am equals
V Am�t� Zt� Zmin���t� � �n
Zmin���t�
�Ho�Zt �K � ����
independently of �max� By Theorem ��� this is the ask�price for the down�and�out call
in a large class of SV�models with �min � �� Let us now look what happens if �min � ��
��
Here we have
V Am�t� Zt� Zmin���t� � �n
Zmin���t� � H
oERZt
���Umin�����min�T�t��
� H�
supnERU���min�T�t��
hf�U� � U
min�����
i� � B�����max��
�min��T�t�
o�
� ERZt
���Umin�����min�T�t��
� H� �U���min�T � t �K
� �� ����
where the last equality follows from ���� � Obviously ���� and hence V Am is equal to the
price of the option in a Black�Scholes model with constant volatility equal to the lower
volatility bound �min� For an explicit formula see again Reimer and Sandmann �����
or Chapter � of Musiela and Rutkowski ����� � It is not dicult to see that V Am is the
ask�price of the option in a large class of SV�models with volatility range ��min�� �
iii Using �vanilla options� to reduce the hedge cost� We now present a nu�
merical example that explains how traded �vanilla options� can be used to reduce the
superhedging cost� We want to hedge a down�and�out call with strike price K � ��� bar�
rier H � ��� and time to maturity � month� assuming that the volatility range is given by
�min � ���� and �max � ���� We moreover assume that we can take arbitrary positions in
a standard call option with K � ��� and time to maturity � month� trading at an implied
volatility of �impl � ���� If we do not take any position in the vanilla call the superhedging
cost for the barrier option is given by the price of the option in a Black�Scholes model
with volatility � � ����� If we add a position of � standard calls to our portfolio� the
hedge cost is given by the sum V Am� � �C�S� � Here V Am
� represents the superhedging
cost of the payo�
f�
�ZT � Z
min���T �
��� �ZT � ����n
Zmin���T �
����o � ��ZT � ���� �
and C�S� denotes the current markety price of the vanilla call� The following table gives
the superreplication cost for � � � and for � � �����Superhedging cost for � � �� ����
Superhedging cost for � � ����� ����
Black�Scholes price for � � ������ ����
We see that by using a static position in the vanilla call we can achieve a drastic reduction
of our hedge cost� Our superhedging price is now much closer to the Black�Scholes price
for a �reasonable� input volatility of � � ������ Of course in our situation one should
choose � so that the superhedging cost of the portfolio is minimized� This idea is developed
systematically in Avellaneda and Paras ����� �
A Appendix
A�� Exchanging conditional expectation and essential supremum
The following proposition� which is adapted from El Karoui ����� � justi�es the exchange
of essential supremum and conditional expectation carried out at various places through�
out the paper�
��
Proposition A��� Consider two stopping times � � � on a �ltered probability space
���F � �Ft � P � Let �ft t�� denote some adapted and RCLL�stochastic process� which is
bounded below� Then we have for two points in time � � s � t � �
ess sup�E�f�
Fs� � � � F� ��� � E�ess sup
�E�f�
Ft� � � � F� ��� Fs� � �A��
Proof� We may write the lhs of �A�� as ess sup�E�E�f� jFt�
Fs� � � � F� ��� � Hence
the lhs of �A�� is a priori smaller than the rhs� De�ne for � � F� �� the Ft�measurable
random variable Y � by Y � �� E�f� jFt�� To show equality in �A�� we have to show that
there is a sequence ���n n�N such that
Y ��n converges monotonically to ess sup�Y � � � � F� ��
�� �A��
in that case the claim follows by monotone integration� It is well�known from general
properties of the essential supremum that there exists a countable sequence ��n n�N such
that ess sup�Y � � � � F� ��
�� sup fY �n � n � Ng � Hence a sequence ���n n�N with �A��
exists� if M� � �� fY � � � � F� ��g is sup�stable�
Consider �� and �� from F� �� and de�ne a set A � Ft by A �� fY �� � Y ��g� De�ne
the random variable �max���� �� by �max���� �� � ���A � ���AC � It is easily seen that
�max���� �� is a stopping time� so that �max���� �� � F� �� � Moreover� as A � Ft�
Y �max � E�f�� jFt��A �E�f�� jFt��AC � Y �� Y �� �
Hence Y �� Y �� belongs to M� � � showing that this set of random variables is sup�stable�
A�� Proof of Lemma ����
Let � be an �Ft �stopping time� Then we have for any t� � �
fhMi� � t�g � f� � At�g�i�� FAt�
� Gt� �
where �i follows as � and At� are �Ft �stopping times� Conversely� as hMit� is a �Gt �stopping time we have for any �Gt �stopping time
fA� � t�g � f � hMit�g � GhMit�� Ft� �
which proves that A� is an �Ft �stopping time�
A�� Proof of Lemma ���
De�ne for a given �Gt �stopping time a random time � by � � A� � By Lemma ���� � is
an �Ft �stopping time� Denote by D����T � the two�dimensional Skorohod space� Hypothesis
���i and ���i imply that the densities Gi�n can be written as functions of the trajectories
of X�
Gi�nt � Gi�n �t� �Xs ��s�t for a function Gi�n � ��� T �� D�
���T � � R with
y�� y� � D����T �� y
� � y� on ��� t� � Gi�n�t� y� � Gi�n�t� y� �
��
Now de�ne for given n�� n� an equivalent martingale measure Q �Me by
dQ
dPFT
�� G��n��� T � X G��n� �T � � T � ��T �X � �A��
where � denotes the shift operator on D�� By de�nition � � infft � �� hMit � g� Hence
Q�hMiT � �� � ��� � Q�hMiT � � hMiT � hMi� � �� �A��
� Q�� � T � �hMiT�� � �� � � � � �A��
Now by de�nition of Q it follows that �A�� equals
EPh�f��TgG
��n��� T � X �f�hMiT��T � ��T � �gG��n� �T � � T � ��T �X i�
Conditioning on F�T we get from the strong Markov property that this is equal to
EP��f��TgG
��n��� T � X EPX�
�G��n��T � � T � X � hMiT��T � �
� �� �A��
Now� using that G��n� is a martingale and that hMit is increasing� we get
EPX�
�G��n��T � � T � X � hMiT��T � �
�� EP
X�
�G��n��T � X � hMiT��T � �
�� EP
X�
�G��n��T � X � hMiT � �
��
Moreover we may obviously replace G��n��� T � X by G��n��T � X in �A�� � Hence
Q�hMiT � �� � ��
� � EP��f��TgG
��n��T � X Q��nX�
�hMiT � ���
�A��
Fix some � �� Choose n� large enough so that Q��n� �� � T � � �� and choose
K �� R � ��� �� with Q��n� �Xt �� K for some t � ��� T �� � �� Now choose �nally for
this set K some n� such that
Q��n�x �hMiT � �� � �� � for all x � K �
This is possible by hypothesises ��iii and ��iii � Hence the rhs of �A�� is minorized by
EP��f��Tg�fX��KgG
��n��T � X Q��n�X�
�hMiT � ���
� ��� � Q��n���� � T � �X� � K
�� ��� �
���Q��n� �� � T ��Q��n�
�Xt �� K for some t � ��� T �
� �� ��� � �� � � �
� ��� �
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